Relations between cone-parameter Lévy processes and convolution semigroups (Q1884811)

The marginal distribution of a Lévy process \((X_t)_{t\in{\mathbb R}_+}\) with values in \({\mathbb R}^d\) forms a convolution semigroup \((\mu_t)_{t\in{\mathbb R}_+}\). Furthermore, the law of the Lévy process is uniquely determined by the convolution semigroup and even by a single measure \(\mu_{t_0}\), \(t_0>0\). They are also stable under subordination. The authors investigate the question which of these properties hold also for Lévy processes \((X_t)_{t\in K}\) indexed by a cone (e.g., \(K={\mathbb R}^N_+\) or \(K=\) the cone of nonnegative definite symmetric \(d\times d\) matrices). It is shown that a cone-parameter Lévy process induces a cone-parameter convolution semigroup. But the converse is not true, there exist cone-parameter convolution semigroups which are not generated by a cone-parameter Lévy process. The authors show that a cone-parameter convolution semigroup is generated by a cone-parameter Lévy process, if it is purely non-Gaussian or if the cone is isomorphic to \({\mathbb R}^n_+\).